My joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” has now been accepted for publication in Journal of Physics A: Mathematical and Theoretical. The arXiv version is arXiv:1412.2719 [math-ph].

I am very happy about this as it is my first joint paper to be published. The paper presents some novel and interesting ideas on how to geometrically formulate higher order mechanics, hopefully our expected applications will be realised.

One interesting possible application, as pointed out by one of the referees, is computational anatomy; this is the quantitative analysis of variability of biological shape. There has been some applications of higher derivative mechanics via optimal control theory to this discipline [1].

We were not thinking of such applications in the biomedical sciences when writing this paper. For me, the main motivation for higher order mechanics is as a toy model for higher order field theories and these arise as effective field theories in various contexts. It is amazing that these ideas may find some use in ‘more down to Earth’ applications. However, we will have to wait and see just how the applications pan out.

How many times have you heard someone say ”I can’t do maths”? Chances are you’ve said it yourself.

du Sautoy talking to the BBC

In all honesty I find myself thinking the above at least twice a day.

Genes or hard work
I am not an expert in how genes play a role in our intelligence, but for sure they do. That said, no-one is born an expert in mathematics and it takes a lot of hard work. Like everything in life, becoming proficient in mathematics to the level you set yourself is about perseverance and the willingness to struggle with things until you have mastered them.

It has now been 99 years, to the day (20/03/2015) since Einstein published his original summary of general relativity [1].

Before that he had published some incomplete works that have the wrong field equation, but the key ideas were in place by 1914. The core idea is that space-time is dynamical and interacts with the matter and energy.

It is hard to believe that this theory of gravity has stood the test of time so well. We know for various reasons that general relativity cannot be the complete picture, but nature just refuses to give us hints on what could be the more complete theory.

I gave a talk the other day based on our recent work on graded bundles in the category of Lie groupoids. Anyway, as part of the motivation I drew the audiences attention to two quotes…

“Mathematics is written for mathematicians.” Copernicus

“For the things of this world cannot be made known without a knowledge of mathematics.” Roger Bacon

They show the two different sides of mathematics; mathematics motivated by mathematics and mathematics motivated by applications. I think one should to some extent sit in the middle here, but ultimately it is nice when mathematics has something to to with the real world, even if that connection is somewhat loose.

My real motivation for these two specific quotes was that Copernicus was Polish and Bacon English!

In collaboration with K. Grabowska and J. Grabowski, we have examined the finite versions of weighted algebroids which we christened ‘weighted Lie groupoids’.

Groupoids capture the notion of a symmetry that cannot be captured by groups alone. Very loosely, a groupoid is a group for which you cannot compose all the elements, a given element can only be composed with certain others. In a group you can compose everything.

Groups in the category of smooth manifolds are known as Lie groups and similarly groupoids in the category of smooth manifolds are Lie groupoids.

It is well-known every Lie groupoid can be ‘differentiated’ to obtain a Lie algebroid, in complete analogy with the Lie groups and Lie algebras. The ‘integration’ is a little more complicated and not all Lie algebroids can be globally integrated to a Lie groupoid. Recall that for Lie algebroids we can always integrate them to a Lie group.

Previously we defined the notion of a weighted Lie algebroids (and applied this to mechanics) as a Lie algebroid with a compatible grading. A little more technically we have Lie algebroids in the category of graded bundles. The question of what such things integrate to is addressed in our latest paper [1].

Lie groupoids in the category of graded bundles
The question we looked at was not quite the integration of weighted Lie algebroids as Lie algebroids, but rather what extra structure do the associated Lie groupoids inherit?

We show that a very natural definition of a weighted Lie groupoid follows as a Lie groupoid with a compatible homogeneity structure, that is a smooth action of the multiplicative monoid of reals. Via the work of Grabowski and Rotkiewicz [2] we know that any homogeneity structure leads to a N-gradation of the manifold in question; and so what they call a graded bundle.

The only question was what should this compatibility condition between the groupoid structure and the homogeneity structure be? It turns out that, rather naturally, that the condition is that the action of the homogeneity structure for a given real number be a morphism of Lie groupoids. Thus, we can think of a weighted Lie groupoid as a Lie groupoid in the category of graded bundles.

I will remark that weighted Lie groupoids are a nice higher order generalisation of VB-groupoids, which are Lie groupoids in the category of vector bundles. These objects have been the subject of recent papers exploring the Lie theory and application to the theory of Lie groupoid representations. I direct the interested reader to the references listed in the preprint for details.

Some further structures
Following our intuition here that weighted versions of our favourite geometric objects are just those objects with a compatible homogeneity structure in [1] we also studied weighted Poisson-Lie groupoids, weighted Lie bi-algebras and weighted Courant algebroids. The classical theory here seems to be pushed through to the weighted case with relative ease.

Contact and Jacobi groupoids
The notion of a weighted symplectic groupoid is clear: it is just a weighted Poisson groupoid with an invertible Poisson, thus symplectic, structure. By replacing the homogeneity structure, i.e. an action of the monoid of multiplicative reals, with a smooth action of its subgroup of real numbers without zero one obtains a principal $latex\mathbb{R}^{\times}$-bundle in the category of symplectic (in general Poisson) groupoids. Following the ideas of [3] this will give us the ‘proper’ definition of a contact (Jacobi) groupoid. We will shortly be presenting details of this, so watch this space.

Prof Beate Heinemann, from the Atlas experiment at CERN had said that they may detect supersymmetric particles as early as this summer. But what if they don’t?

What if nature does not realise supersymmetry? Has my interest in supermathematics been a waste of time?

Superysmmetry

We hope that we’re just now at this threshold that we’re finding another world, like antimatter for instance. We found antimatter in the beginning of the last century. Maybe we’ll find now supersymmetric matter

Prof Beate Heinemann [1]

In nature there are two families of particles. The bosons, like the photon and the fermions, like the electron. Bosons are ‘friendly’ particles and they are quite happy to share the same quantum state. Fermions are the complete opposite, they are more like hermits and just won’t share the same quantum state. In the standard model of particle physics the force carriers are bosons and matter particles are fermions. The example here is the photon which is related to the electromagnetic force. On the other side we have the quarks that make up the neutron & proton and the electron, all these are fermions and together they form atoms.

Supersymmetry is an amazing non-classical symmetry that relates bosons and fermions. That is there are situations for which bosons and fermions can be treated equally. Again note the very different ‘lifestyle’ of these two families. If supersymmetry is realised in nature then every boson will have a fermionic partner and vice versa. In one swoop the known fundamental particles of nature are (at least) doubled! Moreover, the distinction between matter and forces becomes blurred!

A little mathematics
Without details, the theory of bosons requires the so called Canonical Commutation Relation or CCR. Basically it is given by

Here x ‘hat’ is interpreted as the position operator and p ‘hat’ the momentum. The right hand side of this equation is a physical constant called Planck’s constant (multiplied by the complex unit, but this is inessential). The above equation really is the basis of all quantum mechanics.

The classical limit is understood as setting the right hand side to zero. Doing so we ‘remove the hat’ and get

$latex xp- px =0 $.

Thus, the classical theory of bosons does not require anything beyond (maybe complex) numbers. Importantly, the order of the multiplication does not matter here at all, just think of standard multiplication of real numbers.

The situation for fermions is a little more interesting. Here we have the so called Canonical Anticommutation Relations or CAR,

Again these operators have an interpretation as position and momentum, in a more generalised setting. Note the difference in the sign here, this is vital. Again we can take a classical limit resulting in

$latex \psi \pi + \pi \psi =0$.

But hang on. This means that we cannot interpret this classical limit in terms of standard numbers. Well, unless we just set everything to zero. Really we have taken a quasi-classical limit and realise that the description of fermions in this limit require us to consider ‘numbers’ that anticommute; that is ab = -ba. Note this means that aa= -aa =0. Thus we have nilpotent ‘numbers’, that is non-zero ‘numbers’ that square to zero. This is odd indeed.

Supermathematics and supergeometry
In short, supermathematics is all about the algebra, calculus and geometry one can do when including these anticommuting ‘numbers’. The history of such things can be traced back to Grassmann in 1844, pre-dating the applications in physics. Grassmann’s interests were in linear algebra. These odd ‘numbers’ (really the generators of) are usually referred to as Grassmann variables and the algebra they form a Grassmann algebra.

One of my interests is in doing geometry with such odd variables, this is well established and a respectable area of research, if not very well represented. Loosely, think about simple coordinate geometry in high school, but now we include these odd numbers in our description. I will only reference the original paper here [2], noting that many other works evolved from this including some very readable books.

What if no supersymmetry in nature?
This would not mean the end of research into supermathematics and its applications in both physics & mathematics.

From a physics perspective supersymmetry is a powerful symmetry that can vastly simplify many calculations. There is an industry here that works on using supersymmertic results and applying them to the non-supersymmetric case. This I cannot see simply ending if supersymmetry is not realised in nature, it could be viewed as a powerful mathematical trick. In fact, similar tricks are already mainstream in physics in the context of quantising classical gauge theories, like the Yang-Mills theory that describes the strong force. These methods come under the title of BRST-BV (after the guys who first discovered it). Maybe I can say more about this another time.

From a mathematics point of view supergeometry pushes what we know as geometry. It gives us a workable stepping stone into the world of noncommutative geometry, which is a whole collections of works devoted to understanding general (usually associative) algebras as the algebra of functions on ‘generalised spaces’. The motivation here also comes from physics by applying quantum theory to space-time and gravity.

Supergeometry has also shed light on classical constructions. For example, the theory of differential forms can be cast neatly in the framework of supermanifolds. Related to this are Lie algebroids and their generalisations, all of which are neatly described in terms of supergeometry [3].

A very famous result here is Witten’s 1982 proof of the Morse inequalities using supersymmetric quantum mechanics [4]. This result started the interest in applying physics to questions in topology, which is now a very popular topic.

In conclusion
Supermathematics has proved to be a useful concept in mathematics with applications in physics beyond just ‘supersymmetry’. The geometry here pushes our classical understanding, provides insight and answers to questions that would not be so readily available in the purely classical setting. Supergeometry, although initially motivated by supersymmetry goes much further than just supersymmetric theories and this is independent of CERN showing us supersymmetry in nature or not.

A plaque will be placed in Swansea’s Grove Place to commemorate the 19th Century scientist Sir William Grove.

Sir William, was the founder of the Swansea Literary and Philosophical Society, and managed to combine a legal career with several important scientific achievements. In particular he anticipated the conservation of energy and was a pioneer of fuel cell technology. He was the first to produce electrical energy by combining hydrogen and oxygen in 1842, a technology that went on to supply water and electricity for space missions.

Just another example of a Welsh person having done great things for the world.

In collaboration with K. Grabowska and J. Grabowski, we have applied the recently discovered notion of a weighted algebroid to mechanics on graded bundles[1].

In our preprint “Higher order mechanics on graded bundles” We present the corresponding Tulczyjew triple for this situation and derive the phase equations from an arbitrary (maybe singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. This is all done essentially in the first order set-up of mechanics on a Lie algebroid subject to vakonomic constraints. The amazing this is that the underlying graded bundle structure gives this whole picture the flavour of higher derivative mechanics. Within this framework we recover classical higher order mechanics, but we can study some more exotic situations.

For example, we geometrically derive the (reduced) higher order Euler-Lagrange equations for invariant higher order Lagrangians on Lie groupoids. To our knowledge, not much work has been done in understanding such systems [2,3]. We hope that the example on Lie groupoids turns out to be useful, maybe in say control theory.

The videos from the Breakthrough Prize in Fundamental Physics Symposium are now available to watch, follow the link below. The symposium was held on the 10th November at Stanford University and co-hosted by UC-San Francisco and UC-Berkeley.

I did not realise this until today, but I share my birthday with John Wallis (23 November 1616 – 28 October 1703). Wallis made contributions to infinitesimal calculus, analytic geometry, algebra and the theory of colliding bodies. He is best known for the infinity symbol $latex \infty$.

Wallis was also a code breaker and used his mathematical skills during the Civil war to decode Royalist messages for the Parliamentarians.

Notes for Linear algebra 2400-FIM1AL

About the author

My name is Andrew Bruce and I am a mathematical physicist who works on the boundaries of physics and modern geometry.

Currently I am an Assistant Professor in the Department of Mathematical Physics and Differential Geometry, IMPAN , Warsaw, Poland.

My research interests are in mathematical physics and modern geometry. In particular I have been interested in supermanifolds, graded manifolds, Lie algebroids and various generalisations of Poisson structures.

I am also an online tutor in physics and mathematics at Pembrokeshire College.